A stopping time is just that: think that you participate in a sequence of trials, the result of trial #i being Xi. You may stop the sequence at any time. Then obviously your decision to stop at timet depends only on the first t values.

Employ a monkey to bang on a typewriter until 30 minutes before it produces the complete text of Hamlet, then invite all your neighbours to see your educated monkey (bet on it producing Hamlet within 30 minutes).

As you can see, a stopping time is feasible, whereas other rules might not be. And, in fact, various theorems about betting are true for a fixed number of trials or for a stopping time (a fixed number is a stopping time!), but not for just any rule. Wald's theorem is probably the most important elementary result for stopping times.